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On the stable ergodicity of Berger–Carrasco’s example

Part of: Ergodic theory Dynamical systems with hyperbolic behavior Smooth dynamical systems: general theory

Published online by Cambridge University Press:  27 September 2018

DAVI OBATA
DAVI OBATA*
Affiliation:
CNRS-Laboratoire de Mathématiques d’Orsay, UMR 8628, Université Paris-Sud 11, Orsay Cedex 91405, France Instituto de Matemática, Universidade Federal do Rio de Janeiro, P.O. Box 68530, 21945-970, Rio de Janeiro, Brazil email davi.obata@math.u-psud.fr
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Abstract

We prove the stable ergodicity of an example of a volume-preserving, partially hyperbolic diffeomorphism introduced by Berger and Carrasco in [Berger and Carrasco. Non-uniformly hyperbolic diffeomorphisms derived from the standard map. Comm. Math. Phys.329 (2014), 239–262]. This example is robustly non-uniformly hyperbolic, with a two-dimensional center; almost every point has both positive and negative Lyapunov exponents along the center direction and does not admit a dominated splitting of the center direction. The main novelty of our proof is that we do not use accessibility.

Keywords

stable ergodicitynon-uniform hyperbolicitysmooth ergodic theory

MSC classification

Primary: 37A25: Ergodicity, mixing, rates of mixing 37D25: Nonuniformly hyperbolic systems (Lyapunov exponents, Pesin theory, etc.)
Secondary: 37C40: Smooth ergodic theory, invariant measures

Information

Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 40 , Issue 4 , April 2020 , pp. 1008 - 1056
Copyright
© Cambridge University Press, 2018

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